How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 5?

Question

Scarlett Allison · Accepted Answer

Volume of largest rectangular box is #125/162# The volume of the rectangular box in the first octant with three faces in the coordinate planes will be #V=f(x,y)=xyz#. As the vertex lies in the plane #x+2y+3z=5#, #z=(5-x-2y)/3# and volume is #V=f(x,y)=1/3xy(5-x-2y)=5/3xy-1/3x^2y-2/3xy^2# Volume will be maximum if #f_x=f_y=0# As #f_x=5/3y-2/3xy-2/3y^2=1/3y(5-2x-2y)=0# which implies #y=0#, #y=5/2-x# ...............(1) and #f_y=5/3x-1/3x^2-4/3xy=1/3x(5-x-4y)=0# ...............(2) Substituting #y=0# in (2) #1/3x(5-x)=0=>#, #x=0#, #x=5# and at #y=5/2-x# #1/3x(5-x-10+4x)=0# i.e. #x(-5+3x)=0=>#, #x=0#, #x=5/3# At #x=0# #y=5/2# and at #x=5/3# #y=5/6# So critical points are #(0,0)#, #(5,0)#, #(0,5/2)# and #(5/3,5/6)# and #z# at #(5/3,5/6)# is #z=(5-5/3-2xx5/6)/3=5/9# and volume of largest rectangular box is #5/3xx5/6xx5/9=125/162#

Emily Ammerman · Answer

undefined To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $x + 2y + 3z = 5$, follow these steps:

1. The box will have one vertex on the plane $x + 2y + 3z = 5$ and the other three vertices in the coordinate planes. This means that the box will be aligned with the coordinate axes.

2. Let $x$, $y$, and $z$ represent the lengths of the sides of the rectangular box.

3. The point $(x, y, z)$ that lies on the plane $x + 2y + 3z = 5$ must satisfy this equation. Substitute the coordinates of the vertex into the equation to find $z$.

4. Since the other three vertices are on the coordinate planes, their coordinates are $(0, 0, 0)$, $(x, 0, 0)$, and $(0, y, 0)$.

5. The volume $V$ of the rectangular box is given by $V = xyz$.

6. Substitute the values of $x$, $y$, and $z$ obtained in steps 3 and 4 into the formula for volume to find the expression for $V$.

7. To find the maximum volume, differentiate the expression for $V$ with respect to $x$, $y$, and $z$, and set the resulting partial derivatives equal to zero.

8. Solve the resulting system of equations to find the values of $x$, $y$, and $z$.

9. Substitute these values into the expression for $V$ to find the maximum volume of the rectangular box.